Maximal width of the separatrix chaotic layer

TL;DR

This paper investigates the maximum possible width of the separatrix chaotic layer in Hamiltonian systems with a separatrix, revealing universal peak behaviors and providing asymptotic formulas that show the layer can be much wider than previously predicted.

Contribution

It introduces a new theoretical approach to determine the absolute maximum width of the separatrix chaotic layer, including asymptotic expressions for low-frequency peaks, improving upon earlier predictions.

Findings

Maximal width exhibits sharp, universal peaks at certain frequencies.

The maximum width can be significantly larger than earlier estimates.

Theoretical predictions are confirmed through numerical simulations.

Abstract

The main goal of the paper is to find the {\it absolute maximum} of the width of the separatrix chaotic layer as function of the frequency of the time-periodic perturbation of a one-dimensional Hamiltonian system possessing a separatrix, which is one of the major unsolved problems in the theory of separatrix chaos. For a given small amplitude of the perturbation, the width is shown to possess sharp peaks in the range from logarithmically small to moderate frequencies. These peaks are universal, being the consequence of the involvement of the nonlinear resonance dynamics into the separatrix chaotic motion. Developing further the approach introduced in the recent paper by Soskin et al. ({\it PRE} {\bf 77}, 036221 (2008)), we derive leading-order asymptotic expressions for the shape of the low-frequency peaks. The maxima of the peaks, including in particular the {\it absolute maximum} of the width, are proportional to the perturbation amplitude times either a logarithmically large factor or a numerical, still typically large, factor, depending on the type of system. Thus, our theory predicts that the maximal width of the chaotic layer may be much larger than that predicted by former theories. The theory is verified in simulations. An application to the facilitation of global chaos onset is discussed.